Remarks on Texture Coefficients of Polycrystals with Improper Crystallite Symmetry

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The orientation distribution function (ODF) in classical texture analysis is defined on the rotation group SO(3). For polycrystalline aggregates with crystallite symmetry defined by a crystallographic point group \(G_{\mathrm{cr}}\) which is not a subgroup of SO(3), the improper group \(G_{\mathrm{cr}}\) is routinely replaced by its proper peer (i.e., a subgroup of SO(3)) in the same Laue class. In this note we examine how the texture coefficients obtained from such a practice are related to their counterparts that pertain to the corresponding ODF defined on the orthogonal group O(3) as it should.

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  1. 1.

    A survey [7, Sect. 3.5] of circa 127,000 inorganic and 156,000 organic crystals show that a vast majority of them have their symmetries described by improper point groups.

  2. 2.

    The discussion is similar for the case where the orientation space is \(\text{O(3)}/G_{\mathrm{cr}}\).

  3. 3.

    See Sect. 2.1. Note that \(T_{f}: \mathscr{M}(X) \rightarrow \mathbb{R}\), as defined by (6), is vaguely continuous for each \(f \in \mathscr{C}(X)\).


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Correspondence to Chi-Sing Man.

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Man, C., Zhao, D. Remarks on Texture Coefficients of Polycrystals with Improper Crystallite Symmetry. J Elast 138, 111–124 (2020) doi:10.1007/s10659-019-09732-0

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  • Crystallographic texture
  • Orthogonal group
  • Texture coefficients
  • Improper point groups
  • Laue class

Mathematics Subject Classification (2010)

  • 74A99
  • 74E10
  • 74E15
  • 74E25
  • 60B05